Bose--Einstein condensation in the Luttinger--Sy model with contact interaction

TL;DR

This paper investigates Bose--Einstein condensation in a one-dimensional Luttinger--Sy model with contact interactions, demonstrating conditions for generalized BEC, the impact of interactions on condensation type, and bounds on particle density.

Contribution

It establishes the occurrence of generalized BEC under specific conditions and shows how contact interactions change the condensation type in the Luttinger--Sy model.

Findings

Generalized BEC occurs almost surely under certain Poisson potential intensities.

Contact interaction shifts the condensation from type-I to type-III.

Particle density in the largest interval is asymptotically bounded for strong interactions.

Abstract

We study bosons on the real line in a Poisson random potential (Luttinger--Sy model) with contact interaction in the thermodynamic limit at absolute zero temperature. We prove that generalized Bose--Einstein condensation (BEC) occurs almost surely if the intensity $\nu_N$ of the Poisson potential satisfies $[\ln (N)]^4/N^{1 - 2\eta} \ll \nu_N \lesssim 1$ for arbitrary $0 < \eta \leq 1/3$. We also show that the contact interaction alters the type of condensation, going from a type-I BEC to a type-III BEC as the strength of this interaction is increased. Furthermore, for sufficiently strong contact interactions and $0 < \eta < 1/6$ we prove that the mean particle density in the largest interval is almost surely bounded asymptotically by $\nu_NN^{3/5+\delta}$ for $\delta > 0$.